function [J grad] = nnCostFunction(nn_params, ...
                                   input_layer_size, ...
                                   hidden_layer_size, ...
                                   num_labels, ...
                                   X, y, lambda)
%NNCOSTFUNCTION Implements the neural network cost function for a two layer
%neural network which performs classification
%   [J grad] = NNCOSTFUNCTON(nn_params, hidden_layer_size, num_labels, ...
%   X, y, lambda) computes the cost and gradient of the neural network. The
%   parameters for the neural network are "unrolled" into the vector
%   nn_params and need to be converted back into the weight matrices. 
% 
%   The returned parameter grad should be a "unrolled" vector of the
%   partial derivatives of the neural network.
%

% Reshape nn_params back into the parameters Theta1 and Theta2, the weight matrices
% for our 2 layer neural network
Theta1 = reshape(nn_params(1:hidden_layer_size * (input_layer_size + 1)), ...
                 hidden_layer_size, (input_layer_size + 1));

Theta2 = reshape(nn_params((1 + (hidden_layer_size * (input_layer_size + 1))):end), ...
                 num_labels, (hidden_layer_size + 1));

% Setup some useful variables
m = size(X, 1);
         
% You need to return the following variables correctly 
J = 0;
Theta1_grad = zeros(size(Theta1));
Theta2_grad = zeros(size(Theta2));

% ====================== YOUR CODE HERE ======================
% Instructions: You should complete the code by working through the
%               following parts.
%
% Part 1: Feedforward the neural network and return the cost in the
%         variable J. After implementing Part 1, you can verify that your
%         cost function computation is correct by verifying the cost
%         computed in ex4.m

% y is a m x 1 vector, so map y into m x num_labels matrix
Y = [];
I = eye(num_labels);
for i = 1 : num_labels,
  indexs = find(y == i);
  Y(indexs, :) = repmat(I(i, :), size(indexs, 1), 1);
end

% 前向传播, 向量化, 正则化
X = [ones(m, 1) X];  % m x input_layer_size + 1
Z2 = X * Theta1';
A2 = sigmoid(Z2);  % m x hideen_layer_size
A2 = [ones(m, 1) A2];  % m x hidden_layer_size + 1
Z3 = A2 * Theta2';
A3 = sigmoid(Z3); % m x num_labels

% 计算 J, 正则化
Theta1_temp = [Theta1(:, 2 : end)];
Theta2_temp = [Theta2(:, 2 : end)];
sqr_Theta1 = Theta1_temp .^ 2;
sqr_Theta2 = Theta2_temp .^ 2;
cost = Y .* log(A3) + (1 - Y) .* log(1 - A3);
J = 1 / -m * sum(sum(cost)) + lambda / (2 * m) * (sum(sum(sqr_Theta1)) + sum(sum(sqr_Theta2)));
%
% Part 2: Implement the backpropagation algorithm to compute the gradients
%         Theta1_grad and Theta2_grad. You should return the partial derivatives of
%         the cost function with respect to Theta1 and Theta2 in Theta1_grad and
%         Theta2_grad, respectively. After implementing Part 2, you can check
%         that your implementation is correct by running checkNNGradients
%
%         Note: The vector y passed into the function is a vector of labels
%               containing values from 1..K. You need to map this vector into a 
%               binary vector of 1's and 0's to be used with the neural network
%               cost function.
%
%         Hint: We recommend implementing backpropagation using a for-loop
%               over the training examples if you are implementing it for the 
%               first time.

%% 反向传播算法, 向量化, 正则化
% 反向计算各层的误差矩阵 delta
delta3 = (A3 - Y); %.* sigmoidGradient(Z3);
delta2 = (delta3 * Theta2)(:, 2 : end) .* sigmoidGradient(Z2);

% 计算各层的连接参数的梯度 theta_grad
Theta1_temp = [zeros(size(Theta1, 1), 1) Theta1(:, 2 : end)];
Theta2_temp = [zeros(size(Theta2, 1), 1) Theta2(:, 2 : end)];
Theta2_grad = 1 / m * delta3' * A2 + lambda / m * Theta2_temp;
Theta1_grad = 1 / m * delta2' * X + lambda / m * Theta1_temp;

%
% Part 3: Implement regularization with the cost function and gradients.
%
%         Hint: You can implement this around the code for
%               backpropagation. That is, you can compute the gradients for
%               the regularization separately and then add them to Theta1_grad
%               and Theta2_grad from Part 2.
%

% -------------------------------------------------------------

% =========================================================================

% Unroll gradients
grad = [Theta1_grad(:); Theta2_grad(:)];


end
